The Push and Pull of Fortune: Variance and Standard Deviation in Games

Have you ever had the feeling that, even with the same "probability" of winning, some games give you more "thrills" than others? It's not just a baseless feeling! Behind it lies a fundamental statistical concept: variance and its sibling, standard deviation. Imagine two trains departing from the same station and heading to the same destination. One goes straight, the other takes many curves and ups and downs. Both arrive, but the travel experience is very different. In the world of gambling, variance tells us exactly this: the "tendency" of results to deviate from the average, from the expected win or loss.

🎢 Different Journeys to the Same Destination

In a previous article, we talked about expected value, which is somewhat like the "final destination" for a game, or how much we expect to win or lose on average in the long run. But what happens "during" the journey? Variance comes into play here.

Consider two games, A and B, that have exactly the same expected value. This means that, if you play them an infinite number of times, in theory, you would end up with the same final balance. Yet, the experience of playing A might be very different from playing B.

• Game A might offer small but frequent wins. You won't get rich, but you won't lose a fortune quickly either. It's a bit like a peaceful journey on a flat road.
• Game B might offer very large but extremely rare wins, and in the meantime, you constantly lose money. It's like a journey full of steep climbs and breathtaking descents, with the hope of an incredible view at the end.

Both games have the same expected value, but the way they get there is radically different. This difference is quantified by variance.

📐 Measuring the "Fluctuations": What is Variance?

In simple terms, variance measures how much the individual results of a game tend to be far from their mean, or expected value. The more scattered and different the results are, the higher the variance. If, on the other hand, the results are all quite similar and close to the mean, the variance will be low.

Imagine playing a game many times and noting each time how much you won or lost.

• If the numbers you noted are all more or less equal, close to the expected value, the variance is low.
• If, however, you see very different numbers, with large wins alternating with large losses (even if the average at the end is the same), the variance is high.

Variance is calculated using a mathematical formula that takes the difference between each result and the expected value, squares it (to eliminate negative numbers and give more weight to large differences), and then averages it. For a set of data $x_1, x_2, \dots, x_n$ with mean $\mu$, the variance $\sigma^2$ is: $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}$ Don't worry too much about the formula; the important thing is to understand the concept behind it.

📏 Standard Deviation: A "More Understandable" Variance

Variance, being calculated by squaring the differences, has a somewhat strange unit of measurement (for example, if you're talking about euros, variance is in "square euros," which doesn't make much sense in the real world). For this reason, standard deviation is often used, which is simply the square root of the variance.

$\sigma = \sqrt{\sigma^2}$

Standard deviation ($\sigma$) is much more intuitive because it has the same unit of measurement as the original results. If we're talking about wins in euros, the standard deviation is in euros. This tells us, in practice, how "typically" a result deviates from the mean.

• A low standard deviation means that the game results are fairly predictable and close to the expected value.
• A high standard deviation means that the results are much more unpredictable and can vary widely from the expected value.

🎲 Blackjack vs. Slot Machines: A Concrete Example

Let's think about Blackjack and Slot Machines. Both games have a negative expected value (otherwise casinos would go bankrupt!), which means that in the long run, you'll tend to lose. But the gaming experience is very different.

• Blackjack (if played with a correct basic strategy) has a relatively contained expected value and a low variance. Wins and losses are often small, and the game tends to be more "slow" and controlled. You won't make big wins in one go, but you won't lose your entire capital in a few minutes either. The fluctuations in your balance will be contained.

• Slot Machines often have an expected value similar to or even less advantageous than Blackjack, but a much higher variance. The probabilities of winning small amounts are relatively high, but the chance of hitting a huge "jackpot" is extremely low. This means you can lose many rounds in a row, only to (and I emphasize may) then make a big win. The fluctuations in your balance will be very wide, with frequent losses and the hope of a win that pays for everything (and maybe more).

Given the same expected value, those who prefer a calmer gaming experience with fewer "thrills" might prefer Blackjack. Those who are looking for that adrenaline rush from the (remote) possibility of a huge win, accepting many more losses in the meantime, might prefer Slots. Variance is the measure of this preference.

⚠ What to Remember

Variance and standard deviation are powerful tools for understanding not only "how much" one can win or lose on average, but also "how" these wins and losses will occur. They tell us whether a game is more stable and predictable or more volatile and unpredictable, even if ultimately the "house" always maintains its advantage (the negative expected value for the player).

Understanding these concepts helps you make more informed choices and manage your expectations in gaming, recognizing that not all games are equal, even if the basic numbers may seem similar.

Draws are independent random events. Historical statistical analysis does not influence future results. Gambling can cause pathological addiction. Play responsibly. 18+ ADM.